It sounds as if you already know a lot more about staining, refinishing, and the characteristics of beech than BrentD, the guy who is running a campaign to financially damage this forum. Why, you even know how to spell beech.
Minwax, and other finish manufacturers make a Pre-Stain Wood Conditioner that partially seals very porous wood to prevent the problem of uneven stain absorption.
https://www.minwax.com/wood-products/preparation/minwax-prestain-wood-conditionerWatco Danish Oil might give a nice tone to your stock without any additional stain, but I like to try a test piece on a scrap of the same wood before I commit to doing the entire stock. If you try six different neutral wood finishes on the same wood, you will get six different appearances. If you can't come up with a scrap, you might do a test in a small area in the barrel channel.